You are given two components, component A and component B, and suppose their lifetimes, in hours, are expo- nential with parameters XA and AB, respectively. Further, assume that their lifetimes are independent. (a) If AA AB = 1/25, how many hours, on average, does the component that lives longer outlive the other component by? (b) If AA 1/25 and AB = 1/20, what is the probability component A outlives component B?
You are given two components, component A and component B, and suppose their lifetimes, in hours, are expo- nential with parameters XA and AB, respectively. Further, assume that their lifetimes are independent. (a) If AA AB = 1/25, how many hours, on average, does the component that lives longer outlive the other component by? (b) If AA 1/25 and AB = 1/20, what is the probability component A outlives component B?
A First Course in Probability (10th Edition)
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![You are given two components, component A and component B, and suppose their lifetimes, in hours, are expo-
nential with parameters XA and AB, respectively. Further, assume that their lifetimes are independent.
(a) If AA AB = 1/25, how many hours, on average, does the component that lives longer outlive the other
component by?
(b) If AA 1/25 and AB = 1/20, what is the probability component A outlives component B?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27e31058-a1cb-4ad6-9c59-ea40849f1a53%2F513af256-c1e6-4070-b651-3695f127cc94%2Fh1lupy9.png&w=3840&q=75)
Transcribed Image Text:You are given two components, component A and component B, and suppose their lifetimes, in hours, are expo-
nential with parameters XA and AB, respectively. Further, assume that their lifetimes are independent.
(a) If AA AB = 1/25, how many hours, on average, does the component that lives longer outlive the other
component by?
(b) If AA 1/25 and AB = 1/20, what is the probability component A outlives component B?
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
Step 1
According to the provided data,
![А
exp
В ~ еxp(23)](https://content.bartleby.com/qna-images/answer/27e31058-a1cb-4ad6-9c59-ea40849f1a53/0be099c6-dc9c-4ef1-b252-b0c10bb2a6d0/jpi9xfi.png)
Step 2
a. The concept of memory less property is used here, that is, if X~exp(λ), then P(X>t+s|X>s) = e-dt. Therefore, even elapsing or expanding ‘s’ unit of time X still follows exponential distribution.
So, on average the component that lives longer compare to other component have still an exponential distribution with rate 1/25. The average is,
![E(X)
1
1/25
= 25 hours](https://content.bartleby.com/qna-images/answer/27e31058-a1cb-4ad6-9c59-ea40849f1a53/0be099c6-dc9c-4ef1-b252-b0c10bb2a6d0/evmuq6p.png)
Step by step
Solved in 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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